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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvposle | Structured version Visualization version GIF version |
Description: Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
Ref | Expression |
---|---|
fdvposlt.d | ⊢ 𝐸 = (𝐶(,)𝐷) |
fdvposlt.a | ⊢ (𝜑 → 𝐴 ∈ 𝐸) |
fdvposlt.b | ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
fdvposlt.f | ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) |
fdvposlt.c | ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) |
fdvposle.le | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
fdvposle.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥)) |
Ref | Expression |
---|---|
fdvposle | ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossicc 12825 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
3 | ioombl 24168 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol) |
5 | fdvposlt.c | . . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) | |
6 | cncff 23503 | . . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ) |
8 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
9 | fdvposlt.d | . . . . . . . 8 ⊢ 𝐸 = (𝐶(,)𝐷) | |
10 | fdvposlt.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
11 | fdvposlt.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
12 | 9, 10, 11 | fct2relem 31870 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
13 | 12 | sselda 3969 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸) |
14 | 8, 13 | ffvelrnd 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
15 | ioossre 12801 | . . . . . . . 8 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
16 | 9, 15 | eqsstri 4003 | . . . . . . 7 ⊢ 𝐸 ⊆ ℝ |
17 | 16, 10 | sseldi 3967 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
18 | 16, 11 | sseldi 3967 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
19 | ax-resscn 10596 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
20 | ssid 3991 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
21 | cncfss 23509 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) | |
22 | 19, 20, 21 | mp2an 690 | . . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ) |
23 | 7, 12 | feqresmpt 6736 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
24 | rescncf 23507 | . . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))) | |
25 | 12, 5, 24 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
26 | 23, 25 | eqeltrrd 2916 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
27 | 22, 26 | sseldi 3967 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
28 | cniccibl 24443 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) | |
29 | 17, 18, 27, 28 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
30 | 2, 4, 14, 29 | iblss 24407 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
31 | 7 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
32 | 2 | sselda 3969 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
33 | 32, 13 | syldan 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
34 | 31, 33 | ffvelrnd 6854 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
35 | fdvposle.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥)) | |
36 | 30, 34, 35 | itgge0 24413 | . . 3 ⊢ (𝜑 → 0 ≤ ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥) |
37 | fdvposle.le | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
38 | fdvposlt.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) | |
39 | fss 6529 | . . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ) | |
40 | 38, 19, 39 | sylancl 588 | . . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ) |
41 | cncfss 23509 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)) | |
42 | 19, 20, 41 | mp2an 690 | . . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ) |
43 | 42, 5 | sseldi 3967 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
44 | 9, 10, 11, 37, 40, 43 | ftc2re 31871 | . . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
45 | 36, 44 | breqtrd 5094 | . 2 ⊢ (𝜑 → 0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴))) |
46 | 38, 11 | ffvelrnd 6854 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
47 | 38, 10 | ffvelrnd 6854 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
48 | 46, 47 | subge0d 11232 | . 2 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴)) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝐵))) |
49 | 45, 48 | mpbid 234 | 1 ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 ≤ cle 10678 − cmin 10872 (,)cioo 12741 [,]cicc 12744 –cn→ccncf 23486 volcvol 24066 𝐿1cibl 24220 ∫citg 24221 D cdv 24463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-symdif 4221 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-ovol 24067 df-vol 24068 df-mbf 24222 df-itg1 24223 df-itg2 24224 df-ibl 24225 df-itg 24226 df-0p 24273 df-limc 24466 df-dv 24467 |
This theorem is referenced by: fdvnegge 31875 |
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